Imagine you have a process that follows a negative binomial distribution:
for each trial conducted, an event either occurs or does it does not,
referred to as "successes" and "failures". The
frequency with which successes occur is variously referred to as the
success fraction, success ratio, success percentage, occurrence frequency,
or probability of occurrence.

If, by experiment, you want to measure the the best estimate of success
fraction is given simply by k / N,
for k successes out of N
trials.

However our confidence in that estimate will be shaped by how many
trials were conducted, and how many successes were observed. The static
member functions negative_binomial_distribution<>::find_lower_bound_on_p
and negative_binomial_distribution<>::find_upper_bound_on_p
allow you to calculate the confidence intervals for your estimate of
the success fraction.

And now for the important part - the bounds themselves. For each
value of alpha, we call find_lower_bound_on_p
and find_upper_bound_on_p
to obtain lower and upper bounds respectively. Note that since we
are calculating a two-sided interval, we must divide the value of
alpha in two. Had we been calculating a single-sided interval, for
example: "Calculate a lower bound so that we are P%
sure that the true occurrence frequency is greater than some value"
then we would not have divided by
two.